1. PROGRAMME 17
REDUCTION FORMULAS

2. Programme 17: Reduction formulas
Generating a reduction formula
Definite integrals
Integrands of the form and

3. Programme 17: Reduction formulas
Generating a reduction formula
Definite integrals
Integrands of the form and

4. Programme 17: Reduction formulas
Generating a reduction formula
Using the integration by parts formula:
it is easily shown that:

5. Programme 17: Reduction formulas
Generating a reduction formula
Writing:
then
can be written as:
This is an example of a reduction formula.

6. Programme 17: Reduction formulas
Generating a reduction formula
Sometimes integration by parts has to be repeated to obtain the reduction formula. For example:

7. Programme 17: Reduction formulas
Generating a reduction formula
Definite integrals
Integrands of the form and

8. Programme 17: Reduction formulas
Definite integrals
When the integral has limits the reduction formula may be simpler. For example:

9. Programme 17: Reduction formulas
Generating a reduction formula
Definite integrals
Integrands of the form and

10. Programme 17: Reduction formulas
Integrands of the form and
The reduction formula for is
and

11. Programme 17: Reduction formulas
Integrands of the form and
the reduction formula for is:
These take interesting forms when evaluated as definite integrals between 0 and /2

12. Programme 17: Reduction formulas
Integrands of the form and
The reduction formulas for are both:
where
If n is even, the formula eventually reduces to I0 = /2
If n is odd the formula eventually reduces to I1 = 1

13. Programme 17: Reduction formulas
Learning outcomes
Integrate by parts and generate a reduction formula
Integrate by parts using a reduction formula
Evaluate integrals with integrands of the form sinnx and cosnx using reduction formulas